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Relation of Factors & Multiples to GCF and LCM

Let’s explore examples of how factors and multiples relate to the GCF and LCM in more detail, and how they can be applied:

Example 1: Finding Common Factors Using GCF

Let’s say \( a = 48 \) and \( b = 60 \).

Step 1: Find GCF

To find the GCF, list the prime factorizations of both numbers:

  • \( 48 = 2^4 \times 3 \)
  • \( 60 = 2^2 \times 3 \times 5 \)

The GCF is the product of the lowest powers of the common prime factors:

  • \( \text{gcf}(48, 60) = 2^2 \times 3 = 12 \)

Step 2: Use GCF to Identify Common Factors

Factors of 12 are \( 1, 2, 3, 4, 6, 12 \), and these are the common factors of 48 and 60.

  • Common Factors of 48 and 60: \( 1, 2, 3, 4, 6, 12 \).

Example 2: Finding Common Multiples Using LCM

Let’s use the same numbers, \( a = 48 \) and \( b = 60 \).

Step 1: Find LCM

For the LCM, take the highest powers of all prime factors in the factorizations:

  • \( 48 = 2^4 \times 3 \)
  • \( 60 = 2^2 \times 3 \times 5 \)

The LCM is the product of the highest powers of all primes:

  • \( \text{lcm}(48, 60) = 2^4 \times 3 \times 5 = 240 \)

Step 2: Use LCM to Identify Common Multiples

The smallest common multiple of 48 and 60 is 240. All other common multiples will be multiples of 240 (e.g., 480, 720, 960, etc.).

  • Common Multiples of 48 and 60: \( 240, 480, 720, 960, \ldots \).

Application Example: Scheduling Problem

Suppose you have two tasks, one that repeats every 48 minutes and another every 60 minutes. You want to know when both tasks will occur simultaneously.

  • The LCM(48, 60) is 240, so both tasks will occur together every 240 minutes (or 4 hours).

Application Example: Dividing Items into Groups

You have 48 apples and 60 oranges. You want to divide them into the largest possible equal groups of apples and oranges, without any leftover.

  • The GCF(48, 60) is 12, so you can divide them into 12 equal groups, each containing 4 apples and 5 oranges.

Key Insights:

  • The GCF helps with division problems where you're trying to break something into the largest possible equal groups.
  • The LCM helps with scheduling or combining events/tasks that happen at different intervals.